How to evaluate this integral to get pi^2/6:

hb1547

[itex]\int_0^\infty \frac{u}{e^u - 1}[/itex]

I know that this integral is [itex]\frac{\pi^2}{6}[/itex], just from having seen it before, but I'm not really sure if I can evaluate it directly to show that.

I know that:

[itex] \zeta(x) = \frac{1}{\Gamma(x)} \int_0^\infty \frac{u^{x-1}}{e^u -1} du [/itex]

Does the value [itex]\frac{\pi^2}{6}[/itex] come from using other methods of showing the result for [itex]\zeta(2)[/itex] and solving the equation, or is that integral another way of evaluating [itex]\zeta(2)[/itex]?

sunjin09

never mind ... my complex variable technique is rusty ...

hb1547

Anyone else have any input?